Abstract
It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.
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References
Bannai E. et al.: Type II codes, even unimodular lattices and invariant rings. IEEE Trans. Inform. Theory 45, 257–269 (1999)
Conway J.H., Sloane N.J.A: A new upper bound for the minimum of an integral lattice of determinant 1. Bull. Amer. Math. Soc. (N.S.) 23, 383–387 (1990)
Conway J.H., Sloane N.J.A.: A note on optimal unimodular lattices. J. Number Theory 72, 357–362 (1998)
Conway J.H., Sloane N.J.A.: Sphere Packing, Lattices and Groups (3rd ed). Springer-Verlag, New York (1999)
Gaulter M.: Minima of odd unimodular lattices in dimension 24m. J. Number Theory 91, 81–91 (2001)
Harada M. et al.: On some self-dual codes and unimodular lattices in dimension 48. European J. Combin. 26, 543–557 (2005)
Harada M., Munemasa A., Venkov B.: Classification of ternary extremal self-dual codes of length 28. Mathematics of Comput 78, 1787–1796 (2009)
G. Nebe, An even unimodular 72-dimensional lattice of minimum 8, J. Reine Angew. Math., (to appear).
Rains E., Sloane N.J.A.: The shadow theory of modular and unimodular lattices, J. Number Theory 73, 359–389 (1998)
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Harada, M., Miezaki, T. An optimal odd unimodular lattice in dimension 72. Arch. Math. 97, 529–533 (2011). https://doi.org/10.1007/s00013-011-0333-3
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DOI: https://doi.org/10.1007/s00013-011-0333-3